507
Progress of Theoretical Physics, Vol. 16, No. 5, November 1956
Helium Capturing Reactions in Stars
Satio HAYAKAWA*, Chushiro HAYASHI**, Mitsuo IMOTO**
and Ken KIKUCHI*** *>
*
Research Institute fur Fundamental Physics, Kyoto University Kyoto
** Department of Physics, Kyoto University, Kyoto
***Department of Physics, Osaka University, Osaka
(Received July 12, 1956)
The generation of 12C, 16() and 20Ne by successive a-capturing proc•sses in the helium core of stars
is calculated under the condition at temperature-10°°K. The rat<s of forming i~C and 20Ne are
found to be much larger than the previous calculations by Salpeter, Opik and Poyle, due to the
resonance levels recently discovered. The production rates of 24Mg to 4°Ca are also <stimated. The
formation of these nuclei is found to be ne~ligible at temperatur<s below 4 X 10S0 K. The luminosity
of such s:ars of globular clusters belonging to the horizontal branch in the H-R diagram that are
supposed to be burning helium is accounted for in terms of the above thermonuclear reactions at the
central temperature of (1.40±0.10) X10S°K and the density of about 10~gcm-~. The lifetime of a
star in this evolutionary stage is estimated roughly as 107 years. In this course of evolution 20Ne is
formed more abundant than others, in disagreement with the average cosmic abundances of 12C, Ill()
and 20Ne. The modification of the abundances after this stage is suggested as nec<ssary.
§ 1. Introduction and summary
The energy generation in stars belonging to the main sequence has been accounted
for in terms of the " carbon-nitrogen cycle »~>.a> and the " proton-proton chain " 2>. Owing
to these nuclear reactions hydrogen is converted into helium, so that the concentration
of helium in the cores of these stars increases, as they evolve. The helium nuclei thus
produced can hardly be converted into heavier nuclei by the reactions involving light nuclei
alone, because the nuclei of mass numbers 5 and 8 are unstable and even stable isotopes
of Li, Be and B return rapidly back to 3He and 4He due to the absorption of protons
which are rich in the core of such stars 3>. However, the synthesis of carbon and heavier
nuclei is regarded as necessary in view of the evolution scheme, as emphasized by a number
of authors. 4 > Some stars belonging to the " giant branch " are believed to undergo such
a nuclear synthesis that 4He nuclei are converted into 12C and heavier ones. The synthesis
of 12C seems to be required also to explain the over-abundance of 12C observed on the
surfaces of R, N and W olf-Rayet stars.
A possibility of such a synthesis process has been suggested by Salpeter5> as to take
place in hydrogen exhausted stars, when their central temperature becomes higher than
*)
Now at Institute for Nuclear Study, Tokyo University
508
I o·~·K.
S. Hayakawa, C. Hayashi, M. Imoto and K. Kikuchi
At such a high temperature an appreciable amount of 8 Be nuclei is formed under
the thermal equilibrium with 4He nuclei :
(I·I)
A
8 Be
nucleus thus formed can be converted into
12C
by capturing a 4He nucleus :
(I· 2)
These processes are supposed to supply a sufficient amount of energy for about I0 7 years
and may explain the existence of the 12C-rich stars. He has further suggested the synthesis
of heavier nuclei as due to successive a-capturing reactions :
12C+ 4 He~ 16 0+r+7.I5
and
Mev,
16 0+ 4He~ 20Ne+r+4.75
Mev.
(I·3)
(I·4)
In those days the knowledge about the properties of these nuclei was rather poor,
so that his arguments were obliged to be qualitative. Since a level of 12C at 7.68 Mev
and some properties of other nuclei were observed, Hoyle6l tried to calculate the reaction
rates of (I· 2) and (I· 3), taking into account the resonance capture. But the lack in
knowledge of the spins and parities of the levels forbade him to deduce reliable reaction
rates, so that he rather predicted them in reference to the mean cosmic abundances of
C and 160. Lately Salpeter71 noticed the role of the 7.68 Mev level, which was supposed
12
to have zero spin and even parity, and proposed a process,
(I. 2')
But its details and results are not reported yet.
In order to obtain a quantitative answer of the synthesis of heavy nuclei, we attempted to calculate the reaction rates of (I·l) to (I· 4), on the basis of recent nuclear dat.a.8l
We also made a rough estimate on the capture of 4He by 20 Ne to 36A.
Based on our
calculated reaction rates and the rates of energy generation, we briefly discus~ed the synthesis of elements as well as the evolution of stars.
In § 2 the general method of treating the thermonuclear reaction is presented, taking
account of the contributions from resonant as well as non-resonant processes. Far a resonance
energy near the Gamow peak, the con'l'entional method of the Gamow peak is modified, while, far
the sufficient separation between the resonance and Gamow peaks, the contributions from these two
are carefully examined. Our formulas are not qualitati'l'ely different from com·entional ones,
but are useful for quantitati'l'e studies. In § 3 the formation of "Be, and the rates of formation of 12C, 160 and 20Ne and also the rates of the energy generation are calculated.
The result is summarized in Table 3 and in Fig. 5. In § 4 the farmation of 24 Mg to 4°Ca
is discussed.
This is found to be negligible at temperatures below 4 X I0 8·K.
In § 5 the
luminosity of such 3tars that undergo the helium burning processes is interpreted in terms
of a polytrope model of stars. For the stars of globular clusters in the horizontal branch in
the H-R diagram, the central temperature and density are found to be about 1.4 X I0 8•K and
I0 3g on- 3• The reason why these figures are much smaller than expeded by Salpetell is due
Helium Capturing Reactions in Stars
509
mainly to the resonance level of 12 C.
The relative abundances of 12C, 160 and 20Ne are
calculated for given temperatures. In contrast to Hoyle 6 J the abundance of 160 is much lower
than that of 20Ne under the above conditions.
This is due to the newly discovered levefl'J of
20 Ne.
The modification of the abvndance after this evolutionary stage is suggested as necessary.
The mean lifetime of the helium burning stars is estimated as about 107 years under the
same condition and this is in agreement with the observed abundance of the stars in the
giant branch. Our result is in essential agreement with that of Obi et af.B! In § 6 our conclusions are summarized and discussions are made on related problems.
§ 2.
G·eneral formulas
Let us assume a gas composed of two kinds of nuclei 1 and 2, each of which is
uniform in space and obeys the Maxwellian distribution of velocities at given temperctture
T. Suppose that these nuclei of respective densities, n 1 and n 2, collide with one another
with reaction cross section rr (E), where E is the kinetic energy in the center of mass
system of the colliding nuclei. Then the reaction rate per unit time and per unit volume
is given by
p
4 n1n 2
Joo (E) exp ( - EjkT) EdE '
(27rp.)lf2(kT) 3/2 orr
(2 ·1)
where k is the Boltzmann constant. p. is the reduced mass and is expressed, in terms of
the masses of the respective nuclei, m1 and m2, as p.=m 1mi (m1 +m2).
The reaction is regarded as to form a compound nucleus. The energy level of the
compound nucleus is measured from the sum of mass energies of the colliding nuclei and
is designated as En. The property of a level is implied in the widths for the absorption
and the subsequent emission,
and
respectively. sl and s2 show the spins of
target and incident particles, respectively. By employing these quantities the cross section
1s expressed as
rm
rr(E)
r?.n,
1r"h2 ~
1
(2Sr+1) (2S2 +1) 2p.E
n
(2]n+1)F.nrin
(2·2)
(E-E1. ) 2 +Fn2/4
The summation is extended over all possible levels corresponding to spin fm which can
consist of any particular orbital angular momentum In, but one needs in most cases to
take only one level into account.
The total width rn is practically the sum of r.n and
J"';n•
The absorption width r, can be expressed by a classical formula as
r.=G. exp ( -2C1) ,
(2·3)
with
(2·4)
and
G,=2D,j1r·
v
(B-V)j (V +E)::::::2D./7r· -Y"BjV.
(2.5)
510
S. Hayakawa, C. Hayashi, M. lmoto and K. Kikuchi
Z 1e and Z 2e are the charges of the respective nuclei and B the height of the Coulomb
barrier between them. D, is the mean level distance for the levels with the same spin
and parity as the absorbing level and V the depth of the potential for one particle against
the other. These two quantities are rather uncertain, but their influence on the final
result is relatively unimportant. We evaluate D, from observed levels and choose V for
the helium capturing process as
(2·6)
where K 0 = 1 X 10-13 em-1 is supposed to be the wave number of an a-particle just inside
the nuclear surface. 9J The largest uncertainty in the cross section arises from the channel
radius R. In reference to various induced reactions we use the expression
(2·7)
with uncertamty of 0.5 X 10-13 em. R"= 1.6 X 10-13 em is chosen from the a-a scattering and R1 is 2.4 X 10- 13 em for 8Be, 3.8 X 10-13 em for 12C, 4.3 X 10-13 em for 160 and
1..3 A11 ' 3 X 10-13 em for 20Ne and heavier nuclei of mass number A1•
Among various quantities intr~duced above, En and Fin are regarded as independent
of energy, because the energy interval under consideration is relatively narrow. Taking
such energy insensitive quantities out of the integral, (2 · 1) is reduced to
where
= 1.27 X 10- 3ZtZ 2 ~ A1A2/
(At +A2)
eri12•
(2·9)
The numerator in the integrand has the so-called Gamow peak at
Eu= (bkT/2) 213 =0.0264 [A1A2 Z/Zi/(A1 +A2 )]116 T 616 Mev,
(2. 10)
and the width of the peak is
ru= (4E0 kT/3) 112 =0.0178 [A1A2 Z/Zl/ (A1+A2)]1 16 T 516 Mev.
(2 ·11)
Here and hereafter T is measured in 108•K and all energies are in Mev, if not specially
mentioned.
In the presence of the resonance denominator in (2 · 8), the integrand may have
another peak at about E=En. In the integration of (2 · 8), therefore, we have to take
at least two peaks into account. One comes from the peak similar to the Gamow peak,
but is modified slightly due to the denominator. The other is the resonance peak that
plays a dpminant role for IEu-Enl <kT$rg• The approximate methods used in respective cases are described in what follows.
Helium Capturing Reactions in Stars
511
Modified Gamow peak or non-resonance peak.
A peak appears at E 0 which is close to
E0 and To are given by
Eq and its width is 7o-
bkT
=
2E0 ~ 12
1
and
To2
2(E0 -E,.)kT+ (E 0 -ErY+F.,2/4
(Eo-E,Y+Fn 2/4
3
4EokT
(2 · 12a)
(2. 12b)
Then the integral is evaluated as
or
I_,;;;:;:- exp( -b/ v"E;; -EofkT)
,.- "To
(E0 -E,.) 2 +f',.2/4
(2 ·13)
.
It is readily seen that E 0 and To approach E0 and Tu respectively for /E0 -E.,/ >T0 ?;kT
~ F,.. In this case the Gamow appro"imation is subject to the following errors :
(2 · 12a1)
(2 ·12b')
and
The contribution from distant levels is of the order of (E0 -E,.) 2/D/ in comparison with
that from the nearest level at En- If the contribution from the non-resonance peak is
important, /E0 -E,./ is far smaller than D., so that the contribution from distant levels
is negligible.
Resonance peak. The integration in this case can be evaluated as
Irc=:::exp( -bjV'En-E,.fkT) ([ (E-E"') 2 +F,.2/4]- 1 dE
0
2 )[--tan
1t:
( =
2
r,, _
-1( 2E,.
)] exp (
r,
~~
-b
E"')
kT
,,~---.
E..
Smce E.,> F"' in most cases, this can be approximated as
(2 ·14)
The contribution from distant levels is also negligible, because the exponential factor becomes extremely small.
In order to compare the contributions from the resonance and the non-resonance peaks,
we consider three cases, according to the magnitude of /E 0 -E.,/.
r,.,
(a)
/E0 -En/ <
7o· Two peaks overlap with one another and ro:::::::'..F../2. Hence
either of (2 · 13) or (2 · 14) gives us essentially the same result.
(b)
< lEo- E,. I:Sro· There is a sharp resonance peak in ;1 broad Gamow peak.
r,.
512
S. Hayakawa, C. Hayashi, M. Imoto and Kikuchi
Since ro~JE0 -E,.J and the exponential factors in (2·13) and (2·14) are of the same
order of magnitude, we have I,.j[..:::::::..f',.ro/JE0 -E,.J 2• Hence the resonance peak represents
an essential part of the integral I.
(c)
ro< IEo-E,.J. Two peaks are separated from one another. The contribution from the non-resonance peak approaches the Gamow approximation, as E,. become~
far from Eg, So that Eo~ Eg and r 0 ~r g• Comparison has to be made with the numerical
ro and IEo-Enl· With increasing JEo-E,.J, In become~ more important
values of
than Ir, because the exponential factor in Ir decreases more rapidly than (E 0 -E,.) - 2
in I,..
In the resonance reaction. the following approximation is worth remarking.
There
r..,
r..,
often occurs either I'in;? I'sn or I'i,. <{ I'sn. Consequently, one may set either I',.= I',,.
or I',.= I',,.. Since the reaction rate is proportional to I',,. !';,./I',., only !'1" or l',n remains
in either of these cases. In the former case the ambiguity due to the a-width is eliminated,
while in the latter case one need not know the radiation width.
Keeping the above remarks in mind, we express the reaction rate as
(2. 15)
The first summation is due to the non-resonance absorption, while the second one due
to the resonance absorption. In almost all cases one has to maintain only one term in
( 2 · 15) . For the nonresonance absorption
( Al +A2 )~ (2],. + 1) I:.. r ... (Eo);o r-~f210-50.401Ko/kT
2.30 X 10-12
P" (2S1 +1)(2S2 +1)' A1A2 "
(E 0 -E,.)
cm- 3
sec-1
(2. 16)
and for the resonance absorption
Pr=
8.16 X 10-12
(2S1 +1) (2S2 +1)
(
A1 +A2 )
A1A2
3122j (2],. +
n
1) I'inl'sn (E,.) y-st2 10 -50.40lEo/kTcm-asec-t
l',.(En)
(2·16 1)
where T is measured in unit of 10 'K. In each part we usually have to leave only one
term.
To this we have to add the effect of screening. According to Salpeter/0> the weak
screening is a good approximation in our case. Then the reaction rate must be multiplied
by exp(-U0/kT). with
8
(2 ·17)
where p is the density in g em -a.
Z and A are the atomic number and atomic weight of
As we are mainly concerned
the main constituent, which is 4He in most of our cases.
Helium Capturing Reactions in Stars
513
with the case in which p ~ 103 g cm- 3 and T~10B'K, -- U0 jkT is of the order of 10-1 •
Therefore, the correction due to the screening .is of the order of 1 0 % and will not
exceed 30%. Since this order of magnitude is implied in the uncertainty in various
quantities as well as in our method of approximation, we will not include this correction
in our numerical results. As soon aa a more accurate calculation becomes possible, the
correction will easily be taken into account.
The rate of energy generation is obtained by multiplying the Q value, Q, by the
reaction rate and by dividing it by density p, as it is usually expressed in unit of erg g-1
sec- 1 :
S=QPfp.
(2·18)
§ 3. Synthesis of 12C, 160 and 20Ne
As pointed out by Salpeter, 5l an
10-10
l0-15
l0-20
0
10
20
Fig. 1.
T in unit of 10s °K
Concentartion of 8 Be in equilibrium with 4He
xn 6 /px 2ne is plotted against Tin unit of I08°K according to the formula (3·2). xn 6 increases rapidly as temperature increases to 3 X l0S°K, but is
nearly constant above 4X l08°K.
appreciable amount of 8Be can be
formed from 4He in the core of a
star, in which both temperature and
density are very high. The concentration of 8Be is of primary importance in a series of the helium
capturing reactions. This together
with the formation of 12C essentially
determines the rates of forming
subsequent nuclei, 160 and 20Ne.
Since an extremely small fraction
of 8Be nuclei formed in a-a reaction
undergoes further a-capture instead
of disintegrating back into 2a under
the conditions under consideration,
the detailed balancing is realized to
a good approximation in the
reactions 2a~Be8 •
Then, the
concentration of 8Be, n n., can be
expressed in terms of that of 4He,
n0 ., on the basis of the statistical equilibrium as*l
nn./ (n 0 .) 2 = (27t6 2 fkT)
312 m~;jmk,·
exp( -cofkT)
= 1.85 X 10-33-4.84/Px r-3/2 ems,
where mne and mn. are the masses of 8 Be and
the Q value for the disintegration of 8Be.
*)
nne/ (nn 6 )
2
(3 ·1)
He respectively, and c 0 = 0.096 Mev. is
4
given by Hoyle6l is smaller than (3 ·1) by factor two.
S. Hayakawa, C. Hayashi, M. Imoto and K. Kikuchi
514
Expressing the concentration in terms of the relative concentration defined by
x=nmjp,
we have
(3 ·2)
This is plotted against T in Fig. 1. xRe increa:;es rapidly as temperature increases to 3 X
108.K.
Above 4 X 108.K xne/ ("H•r is nearly constant. At T= 108.K and p= 103g
cm- 3, Be8 with concentration XBe of about 10-11 can be formed from the assembly of 4He.
If the capture of an a-particle by 'Be is fast enough, an appreciable quantity of 12C can
be generated, as is seen in the following section. The steep dependence on temperature
explains why the synthesis of heavier nuclei via 8Be is negligible in other stars of lower
temperature, for example in the main sequence stars.
The syntheses of 12C, 160 and 20Ne from 8Be thus formed can be discussed separately
because the formation of 24 Mg becomes comparable only above 4 X 108·K, as shown in § 4.
The process which takes part therein is the radiative capture of an a-particle : the particle
emission is energetically impossible and the formation of an electron pair is less probable.
Hence our first task is to estimate the radiation and a-capturing widths, F; and F., respectively. For these reactions the knowledge about energy levels can be obtained from
existing experiments.UJ
T
T.
T
Sp_zn
parity
T-1,(r)
Mev..
l7.22
16.5
+
2,-T-1
15.09
I
11.1
·10.8
7.12
6.91
2+
6.14.
3"
6. 06
o+
7.85
2+
.4il
Z'
7.22-- 7.. 18
6.74
160+«
(o+J
4.95 ±all2
-t2o±aol
'
I
3<¥
!1
E2
4.43
t~ I
I __
I
I
E2
•'
i!
I'
r
I
El
E1
n·--·---------
z•
I
I
1.58'
:i
~
••a
0+
Fig. 2.
0
0
5:6.3±a02
046
I
I
I
(1 )
9.3·
I
I
1.~14 ~e+ci -7.65
parity
ii.-6·-----------
I
9.61
V'278
parity
~1
I
Spin
Meo
9 58
T
________ _._ -- ( T=1) 7.149
--uc+a
13.3
--'iD6"1i~74-:
f-'-"'--"11.-----r
Spin
Met!.
Level scheme of 12C
Fig. 3.
o+
Level scheme of
160
'''Ne
Fig. 4.
!
z•
Level scheme of 20Ne
The level scheme of 12C is shown m Fig. 2. It is by all means certain that the
7.65 Mev level is onl'Y one which contributes to the absorption of an a-particle.
The spin and
parit'Y of this level are assigned as o+ / 2l so that an S wa'Jie a-particle is captured. In calcula-
Helium Capturing Reactions zn Stars
515
ting the reaction rate we are allowed to use the resonance formula, based on the following. For 0.7;$ T ;:S 30 the contribution from resonance peak is larger than that from
the Gamow peak. At T=21 E0 lies in between 7.65 Mev and 9.61 Mev levels. At
the resonance energy, En= 0.28 Mev, the a-width F, (En) is found to be about 1 ev,
more than three order larger than the radiation width. Consequently, rn~r. and F,jF,.
disappears in the formula of the resonance absorption. Thus we have only to evaluate
ri.
Since the spins and parities of both the absorbing level and the ground state are o+,
the transition to the ground state by emitting one photon is forbidden. The radiative
transition 'to the ground state can take place by emitting two or more photons. It is
usual that the internal pair formation is more probable than the two photon emission in
a zero-zero trans1t1on. In competing this process, the radiative cascade transition through
the 4 · 43 Mev level with 2 + has to be taken into accl)unt. In the last case only the
width for the first E2 transition plays a role in the reaction rate. The widths for these
processes are calculated in the following way.
In the conventional formula for such widths the nuclear matrix element is expressed
:in terms of a "nuclear radius", A! 13~. 9l For the o+ -o+ transition the matrix element
is estimated by Schiff.13J In reference to this we estimate ~= 1.1 X 10-13 em. For the
two photon transition we take the 17.22 Mev level with 1- and isotopic spin unity as
an intermediate state and adopt the same value of R0 • For the E 2 transition the nuclear
matrix element is different and so is R0• The analysis of excitation to the first excited
state by high energy electrons results in the E2 width of 10- 2 ev for the transition
from the first to ground states.11>14l Taking only the energy dependence into account
the E2 width under consideration is predicted as about 10- 3 ev. Since the overlap of
nuclear states is considered as poorer in the latter case than in the former case, this may
be an over-estimate.
It may be a reasonable estimate to choose ~=1.3 X 10-13cm.
Consequently the E 2 width is estimated as 7 X 10- 4 ev. These results are summarized
in Table 1.
Table 1.
Process
Radiation and electron emission for
Transition energy (Mev)
12C
Ro(l0-13 em)
Width (Mev)
E2
7.65--->4.43
1.3
r]iJ2= (7±3) x10-1o
Internal pair
7.65-----+0
1.1
lpair=3X1o-u
2r emission
7.65-----+0
1.1
r2T=5 xl0- 12
From Table 1 we conclude that only the E2 transition needs to be taken into consideration. On account of the uncertainty in m, we shall leave m in the final formula.
Now we are able to give the reaction rate for 0.7;$T;$20. This is presented at
an arbitrary value of T in this interval.
r
PBe+ct.·'>€1
=
10-22
( 7 Xr1~-10 ) 6.o8 x y3/2
.
r
-13.91/T
(3 ·3)
5.16
S. Hayakawa, C. Hayashi, M. Imoto ann · K. Kikuchi
This together with (3 ·1) leads us to
pCBe+ 4He-712C)nn. nR•
P(3 4He-712C)
p3tHc(nn.) 3,
(3 ·4)
with
r"'-'
-( X
P2cHc7
)
p,
10 _10
1. 12 x 1 o-54-18.75/T cm6 sec-t.
T3
(3. 5)
The effect of the screening increases the reaction rate, but the correction is within the
uncertainty in r IC-2·
It is interesting to compare our result with Salpeter's. 5)
g cm- 3 (3 · 5) and (2 · 18) gives us
c= (TE2/7 X 10-10 Mev) (p/2.5 X 104)
2
At T=2 and p=2.5 X 104
1.5 X 109 (T/2 X 10 8°K) 18 "58 (xn.) 3 erg g- 1 sec- 1 •
This is about 106 times larger than that given by Sa/peter.
Such a great difference ts due
mainly to the fact that he did not take the effect of the recently discovered resonance level in 12 C
into account. This is one of the most important conclusions obtained in our work.
(ii)
C+ 4He-7160+7.149 Mev.
12
There are several levels of 160 near the zero energy of the a-particle, as shown in
Fig. 3. Among these levels the contributions from 7.12 Mev (1-) and 6.91 Mev (2 +)
levels need to be compared. The E1 radiation from the former level with zero isotopic
spin would have a far greater width than the E 2 and the E1-E3 cascade radiations from
the latter, if the selection rule with regard to the isotopic spin were not taken into account. As the Z component of the isotopic spin is zero in" the system concerned, the
E1 transition is possible only because of the mixing11> of a state with isotopic spin unity.
The probability for the mixing is believed as about one percent,11 ) so that the E1 width
is smaller by about factor several hundred than in the case where such a selection rule is
irrelevant. Even so the E1 width is nearly several times larger than the E2 width. Moreover,
the resonance denominator in ( 2 · 13) is about four times larger for the 2 + level than
for the 1- level, on account of E 0 =0.2Mev. Therefore, levels other than the 7·12
Mev 1- level contribute to the reaction rate no larger than one percent. The E1 radiation width must be larger than 0.1 ev, because the lifetime of the level is observed as
shorter than 8 X 10-15 sec. 11> Since the mixing of the isotopic spin states is regarded to
be smaller than 1 % it will be safe to limit the E1 width as
0.1 ev<I'm < 1 ev.
(3 ·6)
For convenience we tentatively fix the value of
Tm=3.1 X 10- 7 Mev
(3 ·6 1)
in the final formula.
Then the partial wave which contributes to the absorption of an a-particle is a P
wave. The absorption width is calculated for D.:::::4 Mev and R= 5.4 X 10-13 em as
rs,l-1 (Eg) = 1.62 X 10+6 10:"" 20 "12 /Tl/O Mev,
(3. 7)
Helium Capturing Reactions in Stars
517
r
where E9 =0.2 Mev at 108°K. Hence the total width is practically equal to m and is
negligibly small compared with E9 -En~0.278Mev at T=1.4X10 8°K. This allows us
to use the non-resonance approximation (2·13) under the condition E0 -E,.>r9 ?:,kT~rn.
Now we are able to give the reaction ·rate
(3 ·8)
with
- 4.23 x
Pc-+o-
10-13.
r. ,
1o-30.16/T11• (
TtfJ
.m
'3 X 10- 7 Mev
) cm .sec3
(3. 9)
1
.
The reaction rate is much faster than that for 3 · 4He-712C under the stellar conditions
under consideration.
These formulas are "alid for temperatures below 5.2 X 108oK, at which E 0 lies at the midst
between the 7.12 Me'Y le'Yel and the 8.6 Me'Y le'Yel.
Ab011e this temperature the contribution
from the latter le'Yel has to be taken into account.
A recent experimene 5> has revealed a level scheme of 20Ne different from that given
in reference 11. The new level scheme is shown in Fig. 4. The most important le'Yel is
the 4.95 Me'Y one, but its spin and parity are not yet undetermined. We calculated the radiation
width, assuming the spin and parity of the level as o+, 1- or 2+. The result is shown
in Table 2. Although its 'Yalue strongly depends on the assumed spin and parity, the uncertainty
does not come in the final formula, because of the followmg reasons.
Table 2.
Radiation and electron emission widths (ev) for 20Ne (4.95 Mev level)
Assumed spin and parity of 4.95 Mev level
r,
o+
rm
~air}
.T
rK
rli12
1.1
a(~4.2
((10-6
I
Mev level, 1-)
161.4 a (~ground)
I
z+
1.1
a(~.2Mev
level,
1- or 3-)
(~ground)
5.4Xl0-4(~1.6
Mev level)
3.7x1o-2(~1.6 Mev level)
The decay widths of possible modes are estimated for the 4.95 Mev level of 20Ne, assunting its spin
and pairty as o+, r- and 2+, and using Weisskopf's formula 9l
The factor a in widths shows a constant. If the El transition is partially forbidden by the isospin
selection rule, a= 10-2-10-3, while otherwise a= 1.
At 108°K E 9 is estimated as about 0.25 Mev and, consequently, E1,-En=0.05 Mev.
Since this is as small as
there appears a resonance peak. in a broad Gamow peak.
Hence we are allowed to use the resonance approximation, in which only a smaller one
of radiation and absorption .widths appears. The radiation width lies between 10\3 and
ru,
518
S. Hayakawa, C. Hayashi, M. Imoto and K. Kikuchi
1 ev, as seen in Table 2, ·while the absorption width is estimated as smaller than 10_ 1g
ev. The latter is expressed for the spirt ,of the level, l, R= 5.9 X 10-13 em and D~3 Mev·
as
F,(En) = 1.6 X 10-19 ·10-0 "191 <Hll Mev.
(3·10)
Now one sees that the absorptiop. width is smaller than.ri and thus F,F,j(F'.+l',)~
is a very good. approximation.
Using the resonance approximation (2 · 14), the reaction rate is calculated as
r.
(3. 11)
with
+
(21 1)
1.541(1+1)
2.19 X
10- 31 ·10- 10 .~ 81 7'
(3 ° 12)
T3/2
This is rather insensitive: to the value of. l. The reaction rate is much larger than that for
the formation of 160 at
< 2.6. This is due to the recently obser'/led ler~[ at 4 · 95Mer.
Since the spin and parity of the 4 · 95 Me'll {eye{ are unknown yet, this {eye{ might be
inejfectiye to absorb an a-partide, though this is yery unlikely in Yiew of the nuclear structure
of 20Ne. In order to consider such a case as well as . to see .the competition with .other
levels, .we calculate the reaction rates and the rates of energy generation due to 4.20Mev.
and 5.62Mev levels by using the non-resonance formula:
T
Table 3.
Summary for the synthesis of 12C, 160 and 20Ne
Reaction
Q(Mev)
7.278
7.149
Main absorbing level
7.65 Mev (o+)
7.12Mev (1-)
4.95. Mev (?)
E,.(Mev)
0.28
-0.03
0.20
E0 (Mev)
0.146 '['2/3
0.199 '['2/&
0.247 '['2/ 3
IE,.-E0 1 at T=1*>(Mev)
0.13
0.23
b.05
ru(Mev)
0.0419 T5/6
0.0489 T5/s
0.0544 ya!s
resonance at
o.7;:ST;:S30
·non-resonannce at
~esonance.. or
non-resonance
4.746
T;:S5.2
resonance at
0.5;:ST;S6
rx2=(7±3) x10-1o
ri!Jl ,;_ (0.1~1) X10-6
ri=I0~ 9 ~1o-a
r
r,(E0 )=1.6X10+6
r. (E,.) = 1.6 x w-19
3.16 X 1033-30·15/7'''"
(2/+1)
1.541 (1+ 1)
8
(E,.) = 1.3 X 10-a
3.66 X 1018-18·75/7''
?< 10-20•12/7'1/3
X
'['2/3
X 1o-O·l9l (1+1)
/)XHeXO
1.23 X 1015-10·28/7'
X
s(for E=aX (T/1.4)•)
TS/2
27.8
20.0
15.4
9.61 Mev level
at T:2::21
8.6 Mev level
at T:2::5.2
5.62 Mevlevel
at T:2::6
Uncertainty factor from
and
ri
r.
.
Contributions from
other levels
*)
Tis measured in 1080K. a is the value of
E at T=l.4Xl08"K.
Helium Capturing Reactions in Stars
519
x 1o~u.1o~s7.86/T•'•
r )-1.65
- - - - - - - - : c : : : - - - - - cm
(21 + t) ( ·· _ _i _
.
ho~Ne=
1. 541(1+1>
r·
y2n
10 ~6
3
sec~ 1
(3 ·13)
for the 4.20 Mev level;
[',
•
(21+1) (
Po~Ne
1.541(t+l>
10~6
) 8.76X10~ 12 X10~ 37 • 85iT1 1 "
·
·
T213
.
cm3 sec
~1
(3 ·14)
for the 5.62 Mev level. In obtaining (3 ·13) and (3 ·14), we have evaluated the resonance denominator (En-E9)~ 2 in (2·15) with E0:::=..E9 at T=1.4X10soK. land I',
t0-20
3
2
4
5
T in unit of 108"K
Fig. 5.
--'P'--(,_12_C_+:_a:_~_1a_o-"-)-
·"P'-(""3::.a-~'--1-.o2C=.<)_
X10 3, p3
xa 63
p2
xa.xc
P(160+4He~20Ne)
and --'--------:,------'(J2 XHe Xn
The factor 103 in the case of 3·4He~12C is applied for
the central density of our star, 103 g em-s.
1
4
3
2
T in unit of lOB"K
5
6
Fig. 6.
Energy generation per gram per second for the syn
thesis of 12C, 160 and 20Ne.
This is obtained by ~ultiplying 2.63 by the energy
generation rate for the synthesis· of 12C. Further
correction will be made by multiplying ·a correction
factor f shown in Fig. 10.
520
S. Hayakawa, C. Hayashi, M. Imoto and K. Kikuchi
ri.
may be of the order of 1 to 10- 3 ev in both cases. E'llen if we
are unknown, but
make maximum estimates of the reaction rates, the contributions from these [e,els are negligible in
comparison with that from the 4.95 Me'll le'llel, as far as the latter le'llel is ejfecti'l!e for absorption ; at temperatures abo.,e 6 X 1 0 8" K, the 5.6 2 Me., le.,el becomes more important than the
4.95 Me'll level.
Necessary data for the syntheses of 12C, 160 and 20Ne are summariz.ed in Table 3.
In § 5 we shall see that the temperature and density of interest lie about (1.3.-.1.5)
X 108"K and 103-.104 gjcm3, respectively. Under such conditions, the formation rate of
20Ne is much larger -than ·those of 12C and 160 which are comparable, as seen in Fig. 5.
Therefore, the whole series of the element synthesis is essentially determined by the formation rate of 12C, most of 160 formed from 12C being quickly turned into 20Ne. Hence
the abundance of 20Ne should be larger than that of 160, in contrast to the conclusion of
Hoyle. 6> This problem will be studied in more detail in § 5.
The rate of energy generation due to the whole series is expressed as
c= (Q&.~Op:l<>~O+Qc~o Pa~o+Qo~Ne PMNe) p-l
.
=f(Q:l<>~a+Qa~o+Qo~Ne)Psrs~aP
f
Q:lcHC
-1
=
1.64 X 1011-18.7{;/T
ys
"n/ + Qc~o Pn~oXc/Psrs~G + Qo~Ne Po~N• Xo/ P&.~r(Q:l<>~a+Qo~Ke+Qa~o)Xne.2
f0n.p 2,
(3 ·15)
(3 ·16)
Here f is a quantity of the order of unity which varies slowly as Xn,, x0 and "Ne change
in the course of the element synthesis. The temperature dependence of cj[xn6 3p2 is
shown in Fig. 6. The exact value of this quantity will be calculated also in § 5 in
connection with the problem of the element synthesis.
§ 4. Synthesis of 24 Mg to 4°Ca
Mg, 21!Si, 32S, 36A and 4°Ca can be synthesized by capturing a-particles successively
from 20Ne. The formation rates of these nuclei have never been observed. From experiments of neutron scattering, in which excited energies are higher by a few Mev than
those of interest in the capture of a-particles, the average level distance is expected as
about 0.1 Mev. At high temperatures this is smaller than the width of the Gamow peak, so
that the statistical theory is applied to e'l!aluate the absorption cross sections. In order to obtain
the reaction rates applicable ·for a wide range of temperatures, however, more careful examination is required. (2 · 15) indicates that the ratio of the non-resonance to the resonance
contributions is approximately given by
24
Pn/Pr==-r,,rjD2•
Here the Gamow width r 0 is at most ten times the
total width r is practically equal to the radiation
Hence (4 ·1) is estimated as not larger than 10- 2 •
account of the resonance contribution. As energy levels
(4·1)
level distance D::::=.0.1 Mev and the
width that is at most 10- 4 Mev.
Consequently, we ha'l!e only to take
are unknown, the Gamow energy
521
helium Capturing R.eactions in Stars
may be substituted in place of E,.
in the resonance formula.
p~ (27rjp.kT) 31 ~ iJ 2 n 1 n 2 I'.(Ep)
1020
I
exp( -EufkT)
..
~
'i'
bll
'"§
tote
t
,,,''
,t ,
......
,---
~8.0 X 10- 12[
nl n2I'• (Eu) y-af210-oo.4oEuiT. ( 4. 2)
The formation rates· of 24 Mg to
°Ca are calculated by the use of
(4 · 2) and are shown in Table 4
and Fig. 7. Comparing those rates
with the formation rates of 12C,
160 and 20 Ne shown in Fig. 5, we
find that the formation of 24Mg and
heavier elements in the synthesis of
elements by helium capturing processes
can be neglected in a temperature
range T$4 X lOs'K, unless the density
is appreciably lower than 103 g/ cm 3•
I
I
4
I
I
I
I
I
I
10..2°
l0-24 ......uu_ _.__ _.__ _ , _ _ _ _ , ' - - - - - - - - - '
5
4
3
10
2
0
T in unit of 10soK
Fig. 7. Production rates of 24Mg, 2SSi, 32S, 36A and 40Ca
P(l + 4He-72+r)/p2 Xt xn •.
The dotted curve shows P(160+4He-720Ne)fp2 "H• XNe
of 20Ne with the assumed spm o+ of 4.95 Mev level.
Table 4.
Summary for the syntheses of 24Mg to 40Ca
P(l+He-+2)
Reactions
(A1 +A2) /A1A2] 312
E9 (Mev)
r 9 (Mev)
r.tD
Pt+a.->2
Pl+a.->2""'-a X
(T/4) 8
ntnHe
a
$
I
20N.+a
-724Mu
0.280 T 2/ 3 0.059 T 516 108• so-29· 807'-1/8
2.6 X 10-4-43. 927'-1/8
T3/2
7.3 X l0- 33
21.2
24Mu+a
-72BSi
0.331 T 2i 3 o.o629T 5/6 109· 42-33· 417'-1/8
3 . 6 x 10-3-so. loT-1t•
ys12
1.1 x10-~ 5
24.2
28Si+a
-7a2s
0.369 T2/3 o.0665T5/s 1010. 49-37· 557'-1/8
4.o x lo-2-56. 127'-""
TS/2
2.4Xl0- 38
27.1
32S+a
-786A
0.406 T 213 o.o699T5fs 1011· 69-41· 207'-1/3
4 .2 x 10-1-s1. 661'-118
ya12
8.1 X l0- 41
30.1
36 A+a
-74oc,.
0.441 T2/ 3 0.0726T 516 1Q12· 47-44. 707'-1/8
4.0 X 10-aa. 907'-11•
ys12
3.8x10-4a
32.3
Production rates and necessary quantities for them are listed for 24 Mg, 2SSi, 32S, ~ 6 A and 40Ca.
expressed in unit of 10soK.
T is
'522
S. Hayakawa, C. Hayashi, M. Imoto and K. Kikuchi
§ 5. Application to astrophysical problems
The results obtained in the foregoing sections will be applied to estimate the tern·
perature 'in the core of stars belonging to the horizontal branch in the H-R diagr~ and
to calculate the relative abundances of 12C, 160 and 20Ne which are formed in the interior
of these stars.
A. The central temperature. In a recent study, Hoyle and Schwarzschild16> has shown
that stars of gl<?bular clusters with mass slightly larger the solar mass evolve from the
main sequence to the red giant within a time of about 6 X 109 years, increasing their
central temperature. In the core of such stars hydrogen has been converted into helium.
As the temperature in the core increases as high as lOs'K, helium nuclei should begin to
react with themselves, forming carbon.
As soon as this reaction starts, the stars· are
presumed to evolve from right to left along the horizontal branch in the H-R diagram.16>17>
The luminosity of stars in this branch ' is given by
(5. 1)
L= (40--1600) L®
corresponding to the absolute photo-visual magnitude 1---3. This luminosity is to be explained in ~erms of the helium reactions which are assumed to be the main energy sources
of these stars.
In order to derive the luminosity on the basis of the rate of energy generation obtained in § 3, one has to know the structure of the stars. Since the energy generation
occurs only in a small central region on account of its steep temperature dependence,
which is expressed from (3 ·15) as
(5. 2)
with s=40-.26 for T= (l.0-.1.5) X lOs, we can take a polytropic model with index n
for the structure of this region. Then, neglecting the radiation pressure compared with
the gas pressure and also the electron degeneracy, distributions of temperature and density
are expressed in terms of the Emden function {} ( ~) as· follows :18>
T/Ta={}, pjp,.={}", r=a~,
(5 ·3)
a2 = (n + 1) kTc/47rGf1mnP0 ,
and
where G is the constant of gravitation and p. the mean molecular weight of gas.
For
the present, T is expressed in ordinary unit. The total energy liberat10n is given by
(5. 4)
Evaluating the last integral in ( 5 · 4) by apprpximating {} ( ~) for ~ ~ 1 as
1 ~-+-~
c•
n c4 -···=e-1: •;6 { 1+ ( n -1 )~4
{} =1-. +···
6
120
120
72
}
(5 5)
and by extending the integration limit to ~ = co, which produce no appreciable errors for
3n+s~ 1, we find
Helium Capturing Reactions in Stars
k )s'2pst2y<+3/2
L --s0 x He a,/
J \3n+3
--c
c
•
·' 3n+s 2Gt-tmn
523
(5·6)
Noting that (5 · 6) is rather insensitive to the value of n so long as s>n, we take
as the appropriate values of n and s
n= 1.5 and s=28,
(5 · 7)
which correspond to the polytropic index for the convective equilibrium and to the temperature 1.4 X 10R"K, re~pectively. Further, expressing Tc in unit of 10R"K and putting
(5·8)
we have from (5 · 6)
The value of a defined by ( 5 · 8) depends on the whole structure of stars, but it is known
to be of the order of unity for stellar model of helium burning stars studied. preliminarily
by Hoyle and Schwarzshild.16) The determination of Tc from (5 · 9) is no_t sensitive to
the value of (a j 11) 3' 2 .,c:'k, f so that we fix it as
(ajp) 312x'iT.{=1
on account ofa;2';1,xn.$1, f-:=.1and2>!1 <:4/3.
(5·10)
Then,wefindfro m (5·1) and{5·9)
T.= (1.40 ± 0.10) X 108 °K
(5 ·11)
for the stars belonging to the horizontal branch. Since the rate of energy generation around
. this temperature depends very steeply on T., T. is determined within a .ra:ther narrow
range, even if other quantities a and Xne are varied in a wide range. For example,
decreases of (a xn}/ p)
by factor 10 and 10 3 results in the increases of T. by 7
percent and 20 percent, respectively.
As has been stated in § 3, the value of Tc obtained above is far smaller than that
by Salpeter. 5l This is very important in discussing the synthesis of elements in the stars,
as shown in what follows.
B.. Abundanc-es of 4He, 12 C, 160 and 20Ne. Since the central temperature is determined
as rather low, the formation of 24Mg and heavier nuclei can be neglected. in discussing
the synthesis of elements in the stars under consideration.
The time variation of relative abundances is expressed by the following set of equations.
sty
dxn./ dt=- 3 (pj M,.) 2 Psa-+o "n• (xn/ +tc?Co/3 +tc'xof4),
(5 ·12a)
dx0 jdt=3 (pjM,.) p
(5 ·12b)
2 3,._. 0
and
Xn0 (xn/-tcx0 ),
d'Xo/dt=3 (pjM,.) 2 Psa-+0 'Xne ( (4/3) tcXo-tc1'Xo),
(5 ·12c)
dxNe/dt=3 (pjM,.) 2 Psa-.o'Xne (5j4)tc'Xo·
(5 ·12d)
Here tc and tc' are dimensionless quantities defined by
Po-..Ne M,.
~Paa-+0 P
(5 ·13)
52'4
'i's
u
S. Hayakawa, C. Hayashi, M. Im.oto and K. Kikuchi
·The values of ate and ate' are represented in Fig. 8 with possible uncertainties. As the temperature specifying te and
te1, we can take the central one on accouqt
of the steep temperature dependence of
reaction rates.
Indeed, the integrand in
(5 · 4) has its maximum at 1-0..:_1- Tj
T0 =1/(3n+s)~1/32;5. Then, we have
te~1. and te 1 ~100, if we adopt the most
probable values represented by solid curves
irt ~ig. 8. At the same temperature (3 · 5)
·leads us to
10'
cr~
.
...
"1:1
bll
~
"1:1
!;j
..
1u•
~
o-• '----:---~----!;3----'4;----';--__:;;~...J
T in unit of .lOs °K
Fig. 8. "'" and "'"'
Dashed curves show their domain of uncertainties
arising fromri and
l indicates the assum~d
spin of 4.95Mev level of 20Ne which has not
,been ·determined yet.
r..
We· ~hall first examine the qualitative
nature of the solutions. of'(5 · 12), assuming that changes of te and te 1 are small in
the course of the element synthesis. For
t < 1/lite 1.-....10 1z sec, . x.He can be regarded as
unity and (5 · 12) are solved a~
(5 ·15)
At t~1/JJte 1 , (5 · 15) shows that we have tex 0 c::=..te1x0 and x0~XNe in order ofmagnitude.
For t?::,1jJJte 1 we can put an approximate relation,
(5 ·16)
on account of the high values of te 1 and te 1jte.
The relation (5 ·16) states that the
number of 160 formed from 12C per unit time is nearly equal to that converted into 20 Ne.
Solving again (5 ·12) with (5 ·16) for 1/JJte1$t< 1/JJ, we have in order of magnitude
(5 ·17)
It is found in this way that the abundance of 160 rem~ins very small compared with
that of 12C and also that of 20Ne except for a short period r::::;ljvte'. This result is in
contradiction with that of Hoyle6 > who discarded the formation of 20Ne and dete~ined
the stellar temperature based on the cosmic abundance of 12C and 160.
A more detailed study of the relative abundances is carried out by the numerical
solution of (5 ·12). For this purpose we eliminate time t from (5 ·12) and obtain a
set of equations for Xa. x0 and XNe against "n•·
dxofdxn.=-
:rn.-ICXo
xn/+ (te/3)x0 + (te'/4)xo
(5 · 18a)
(5. 18b)
helium Capturing R~:actions in Stars
525
and
(5 · 18c)
Fig. 9a
Fig. 9b
Fig. 9c
(J. .s-x,,~-
Fig. 9.
Relative abundances of 12C,
I
a
16()
ro
and 20Ne
" and "' are chosen respectively as follows :
IC
"'
5
250
250
b
1
c
d
0.2
0.2
Fig. 9d
250
0.1
The solutions of these equations are given in
Fig. 9 for a number of given sets of JC and
I I
1•
I I
JC
It is found in the interesting cases, JC 1 ~
I I
I I
I I
1 and JC 1 ~ JC, that the final abundance of 160
I I
.4.0
:
I
is very small compared with those of 12C and
20 Ne and that the final ratio x0 /xN• is less
I
3. 0
I
than 0.6 percent for JC:;::;0.6.
In the same
I
I
interesting case, the factor (3 · 16) in the
I
I
I
total energy generation is also plotted against
2. 0
Xn6 in Fig. 10.
\ ..--·· ·As the helium reactions proceed with
l. 0
_.//'',,,, -----··-···".;:::;···········---.....___
Xn6 decreasing, the central temperature of the
/
-~--:.::/
/
~< -;-0~:rstars must increase in order to give the
0; oo.··~
a5
--xn.
J.o
nearly constant luminosity of the horizontal
T in unit of wsoK
branch. However, (5 · 9) together with Fig.
Fig. 10. Correction factor for the energy
10 show& that the required increase in terngeneration
perature is rather small unless ~n. becomes
very small, say, ten percent. Therefore, results obtained above are not altered essentially
if the variations of /C and JC 1 are taken into account in solving (5 ·18).
f
5.0
"
\
526
S. Hayak.awa, C. Hayashi, M. Imoto and K. Kikuchi
At T~1.4 X 108'K, most of helium nuclei are found to be converted into 20 Ne, in defnite
disagreement with the observed cosmic abundances. It will not be correct, however, if one thinks
that the cosmic abundances of 12 C, 160 and 20Ne are accounted for merely in terms of the helium
capturing processes. For the elements thus formed will be converted into heavier ones in the later
stages of stellar evolution.
For example, 20 Ne(a, r) 24Mg and 20Ne(p, r)21Na will be
important processes to eliminate abundant 20Ne, if there occurs some mixing of the chemical
compos1t1on. 21Na (f9+ JJ) 21Ne and 21 Ne (a, n) 24Mg are considered as sources of building up heavier elements by W.A. Fowler et al/9> On the other hand, if there is no
mixing 20Ne+ 20Ne~4°Ca+r will be expected at the temperature <:-10 9 °K.
The heavy
elements thus built up may be broken up through spallation reactions caused by energetic
particles, when stellar matter is scattered into the interstellar space. 20>2l>
§ 6. Conclusions
We have calculated the rates of thermonuclear reactions caused by successive helium
capturing processes in the helium core. The rates obtained by us are much faster than' those
obtained previously by Salpeter 5> and Hoyle. 6> This is mainly because the resonance levels are
discovered for 12 C and 20 Ne. Our results are in essential agreement with those calculated by Obi
et a[.8 >
If the rate of energy generation is applied to interpret the luminosity of the stars
belonging to the horizontal giant branch in the H-R diagram, the central temperature is
estimated as (1.4 ± 0.10) X 108'K for the density of about 103g cm- 3• At about this temperature the formation of 24Mg and heavier nuclei can be neglected and there occurs a series of
syntheses from h#ium to neon. In this series the formation of 12 C from three 4 He nuclei is
the slowest reaction, so that this determines the reaction rate of this series. The lifetime for
the giant stars to reach the stage of R.R. Lyrae type variable stars is estimated as
about 107 years.
In this course of evolution 20 Ne is formed more abundant than others•
.This suggests that the element abundances determined by the helium capturing processes are
considerably modified in later stages of evolution.
Our result can also be applied for the primeval synthesis of elements in the a-/9-r theory.
The most serious difficulty in their theory lies in the point that the successive formation of
heavier elements is locked at unstable nuclei with mass numbers five and eight. However,
the large reaction rate of forming 12C from three a-particles may remove such a difficulty.
This problem will be discussed in a separate paper by one of us (C. H.) and his collaborators.
We should like to express our sincerest gratitude to Professors T. Hatanak.a, Z.
Hitotuyanagi and M. Tak.etani; their view of the stellar evolution motivated us to work
in this subject. We are also grateful to Professor S. Nakamura and his collaborators,
with whom we have had constant discussions on the same problem.
Helium Capturing Reactions in Stars
527
References
1)
2)
H. Bethe, Phys. Rev. 55 (1939), 434; Ap. J. 92 (1940), 118.
H. Bethe and C. R. Critchfield, Phys. Rev. 54 (1938), 248.
E. E. Sal peter, Phys. Rev. 88 (1952), 547.
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
W. A. Fowler, Proceedings of the Symposium on Nuclear Processes in Celestial Objects, Liege
(1954), 88.
E. E. Sal peter, Phys. Rev. 97 (1955), 1237.
M. Taketani, T. Ha~anaka and S. Obi, Prog. Theor. Phys. 15 (1956), 89; earlier papers are
cited there.
E. E. Salpeter, Ap. J. 115 (1952), 326.
E. E. Salpeter, Ann. Rev. Nucl. Sci. 2 (1953), 4.
F. Hoyle, Ap. J. Supple. 1, No. 2 (1954), 121.
E. E. Salpeter, Phys. Rev. 98 (1955), 1183. ·The authors are indebted to Professor Salpeter who
let them know his calculation of the radiation widths concerning the helium capturing proCESSEs.
The same attempt was carried out by S. Obi and his collaborators. Prog. Theor. Phys. 16 (1956),
389.
With these authors we have made a close contact and enjoyed stimulating discussions.
]. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (1952).
E. E. Salpeter, Aust. J. Phys. 7 (1954), 373.
F. Ajzenberg and T. Lauritzen, Rev. Mod. Phys. 27 (1955), 77.
V. K. Rasmussen, D. W. Miller and M. B. Sampson, Phys. Rev. 100 (1955), 181.
]. Seed, Phil. Mag. 46 (1955), 100. Harries, Proc. Phys. Soc. 67A (1954), 153.
L. I. Schiff, Phys. Rev. 98 (1955), 1281.
D. G. Ravenhall, private communication. D2vons, Manning and Towle, Proc. Phys. Soc. 69A, No.
434A, 173. Others cited in ref. 11).
R. G. Freemantle, D. F. Prouse, A. Hossian and J. Rotblat, Phys. Rev. 96 (1955), 1270.
G. Shrank and G. K. O'Neill, Bull. Am. Phys. Soc. 1(1956), No. 1, 29.
F. Hoyle and M. Schwarzschild, Ap. J. Supple. 2, No. 2 (1955).
T. Hatanaka and S. Obi,. private communication.
See, for example: S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Chicago
(1939).
W. A. Fowler, G. R. Burbidge and E. M. Burbidge, Ap. J. 122 (1955), 271.
W. A. Fowler, G. R. Burbidge and E. M. Burbidge, Ap. J. Suppl. No. 17 (1955) 167.
S. Hayakawa, Prog. Theor. Phys. 15 (1956), 111.